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Chapter 5: Problem 22

Find each product. When possible, write down only the answer. $$(4 y+w)(w-2 y)$$

Short Answer

Expert verified
w^2 - 8y^2 + 2yw

Step by step solution

01

Apply the Distributive Property

First, distribute each term in the first binomial \( (4y + w) \) to each term in the second binomial \((w - 2y)\). This gives us: \[ (4y)(w) + (4y)(-2y) + (w)(w) + (w)(-2y) \]
02

Simplify Each Term

Now, simplify each product: \[ (4y)(w) = 4yw \] \[ (4y)(-2y) = -8y^2 \] \[ (w)(w) = w^2 \] \[ (w)(-2y) = -2yw \]
03

Combine Like Terms

Combine the like terms \((4yw \text{ and } -2yw) \): \[ 4yw - 2yw = 2yw \] So the expression becomes: \[ w^2 - 8y^2 + 2yw \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Distributive Property
To multiply two binomials together, we use a very important rule in algebra called the distributive property. This property says that you need to multiply each term in the first binomial with every term in the second binomial. It's like ensuring you cover all possible combinations.

For example, when you have \( (4y + w)(w - 2y) \), you would multiply every term in the first binomial (4y and w) with every term in the second binomial (w and -2y). Think of this like distributing each term to all the others:
\[ (4y)(w) + (4y)(-2y) + (w)(w) + (w)(-2y) \]
Once you distribute, you just do the multiplication for each of the separate parts.
Like Terms
After you've done the multiplication using the distributive property, the next step is to combine like terms. Like terms are terms that have the same variables raised to the same powers. For instance, in the expression \( 4yw - 2yw \), both parts have the variable 'yw', making them like terms.

You sum or subtract the coefficients (the numbers in front) of these terms to simplify the expression. In our case, combining \( 4yw \text{ and } -2yw \) results in \( 2yw \). So, always look for terms that share the same letters and exponents to simplify your algebraic expressions.
Algebraic Expressions
Algebraic expressions are combinations of numbers, variables, and operations (like addition and multiplication). They are like phrases in algebra that can represent numbers and relationships between them. Understanding how to manipulate these expressions is key to solving many problems in algebra.

For instance, the expression we worked with, \( (4y + w)(w - 2y) \), involved transforming and simplifying it step by step until we got the final expression: \[ w^2 - 8y^2 + 2yw \].

The simplified expression represents a combination of various terms into a more compact form. By mastering these techniques, you can tackle more complex algebraic problems with confidence.

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